Shadowing and Hyperbolicity for Endomorphisms of Locally Compact Groups

Abstract

We study shadowing, with respect to the left uniformity, for continuous endomorphisms of Lie groups and totally disconnected locally compact groups. For Lie groups, an endomorphism has shadowing if and only if its differential is hyperbolic, with zero eigenvalues allowed. This includes singular maps outside the classical theory of Anosov endomorphisms. As consequences, positively expansive Lie group endomorphisms are topologically expanding, while on connected semisimple Lie groups the shadowing endomorphisms are precisely the nilpotent ones. On compact connected Lie groups, the nonsingular case agrees with classical Anosov theory. In sharp contrast, every continuous endomorphism of an arbitrary totally disconnected locally compact group has shadowing, without compactness, metrizability or invertibility assumptions; the proof uses Willis' tidy-above decomposition. Consequently, in this category topological expansion and the topologically Anosov property reduce to positive expansiveness and expansiveness, respectively. We also discuss group shifts and revisit Aoki's dense-orbit compactness result without assuming metrizability.